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Geometric Time-Dependent Density Functional Theory

Éric Cancès, Théo Duez, Jari van Gog, Asbjørn Bækgaard Lauritsen, Mathieu Lewin, Julien Toulouse

Abstract

We provide a new formulation of Time-Dependent Density Functional Theory (TDDFT) based on the geometric structure of the set of states constrained to have a fixed density. Orbital-free TDDFT is formulated using a hydrodynamics equation involving a new universal density-to-current functional map. In the corresponding Kohn--Sham equation, the density is reproduced using a non-local operator. Numerical simulations for one-dimensional soft-Coulomb systems are provided.

Geometric Time-Dependent Density Functional Theory

Abstract

We provide a new formulation of Time-Dependent Density Functional Theory (TDDFT) based on the geometric structure of the set of states constrained to have a fixed density. Orbital-free TDDFT is formulated using a hydrodynamics equation involving a new universal density-to-current functional map. In the corresponding Kohn--Sham equation, the density is reproduced using a non-local operator. Numerical simulations for one-dimensional soft-Coulomb systems are provided.
Paper Structure (1 section, 1 theorem, 33 equations, 2 figures)

This paper contains 1 section, 1 theorem, 33 equations, 2 figures.

Table of Contents

  1. Geometric Runge--Gross theorem

Key Result

Theorem 1

Let $\Psi_1(t)$ and $\Psi_2(t)$ be the solutions to the Schrödinger equations with $m=1$ or $m=2$, starting at the same initial condition $\Psi_0$. If $\rho_{\Psi_1(t)}=\rho_{\Psi_2(t)}$ for $t\in [0,T)$, then $W_1(t,\mathbf{r})=W_2(t,\mathbf{r})$ for all $\mathbf{r}\in\mathbb{R}^d$ and all $t\in[0,T)$.

Figures (2)

  • Figure 1: In the Geometric Principle, the velocity $\partial_t\Psi(t)$ is the projection of $-i\widehat{H}(t)\Psi(t)$ onto the tangent space $\mathcal{T}_{\Psi(t)}\mathcal{M}_\rho$. The normal space is given by \ref{['eq:normal_space_Psi']} and one arrives at Eq. \ref{['eq:Schrodinger_geometric']}.
  • Figure 2: Rabi oscillations in a 1D Helium-like atom with soft-Coulomb potential subjected to a time-dependent uniform electric field with amplitude $E=0.00667$ and angular frequency $\omega=0.5336$. Heatmaps of the non-adiabatic contribution $V^\text{na}$ to the exact time-dependent Kohn-Sham potential (left) and of the non-adiabatic geometric correction term $W^\text{na}$ (right), at times $|t-T_{\rm R}/4|\leqslant 2 T_{\rm opt}$ (bottom) and $|t-T_{\rm R}/2|\leqslant 2 T_{\rm opt}$ (top) with $T_{\rm opt}=2\pi/\omega$ the optical period.

Theorems & Definitions (1)

  • Theorem 1: Geometric Runge--Gross